Basics of Fluid Mechanics - The Orange Grove

8 CHAPTER 1. INTRODUCTION
Combining equation (1.9) with equation (1.6) yields
τxy = µ δβ
δt
(1.10)
If the velocity profile is linear between the plate (it will be shown later that it is
consistent with derivations of velocity), then it can be written for small a angel that
δβ
δt
= dU
dy
(1.11)
Materials which obey equation (1.10) referred to as Newtonian fluid. For this kind
of substance
τxy = µ dU
dy
(1.12)
Newtonian fluids are fluids which the ratio is constant. Many fluids fall into this
category such as air, water etc. This approximation is appropriate for many other
fluids but only within some ranges.
Equation (1.9) can be interpreted as momentum in the x direction transfered
into the y direction. Thus, the viscosity is the resistance to the flow (flux) or
the movement. The property of viscosity, which is exhibited by all fluids, is due to
the existence of cohesion and interaction between fluid molecules. These cohesion
and interactions hamper the flux in y–direction. Some referred to shear stress
as viscous flux of x–momentum in the y–direction. The units of shear stress are
the same as flux per time as following
F
A
kg m
sec 2
1
m2
= ˙m U
A
kg
sec
m
sec
1
m2
Thus, the notation of τxy is easier to understand and visualize. In fact, this interpretation
is more suitable to explain the molecular mechanism of the viscosity. The
units of absolute viscosity are [N sec/m 2 ].
Example 1.1:
A space of 1 [cm] width between two large plane surfaces is filled with glycerine.
Calculate the force that is required to drag a very thin plate of 1 [m 2 ] at a speed of
0.5 m/sec. It can be assumed that the plates remains in equiledistance from each
other and steady state is achived instanly.
SOLUTION
Assuming Newtonian flow, the following can be written (see equation (1.6))
F =
A µU
h
∼ 1 × 1.069 × 0.5
0.01
= 53.45[N]